It is provably incorrect, since there are people who love others while not loving themselves. What your LCM said is one of those glib sayings that may sound sensible and comforting, but which actually mean nothing, since they are patently not true.

And "love" is a really nebulous concept. The term means different things for different people. In addition, there is loving feelings and loving behaviour, and those do not have to be connected at all - you can have either one of the two without the other. So no, I disagree with her in several different ways, and I think that she really needs to think about what she wants to say, instead of just resorting to trite sayings that are basically empty of real meaning.